Abstract
In this paper, we study through the sub- supersolutions method the existence results of positive weak solution for the singular nonlinear system
{-div[vertical bar x vertical bar(-ap) vertical bar del u vertical bar(p-2) del u] = gimel vertical bar x vertical bar(-(alpha+1)p+c) u(alpha) in Omega
u > k in Omega
u = K on partial derivative Omega.
where Omega subset of R-n is a bounded domain with smooth boundary partial derivative Omega, 0 epsilon Omega, 0 <= a < n-p/p, 1 < p < n, gimel is a positive parameter, c > 0; 0 < alpha < p - 1, k epsilon [ 0, infinity) : For k epsilon [ 0, infinity), we prove that the above system have a positive weak solution when gimel >= gimel*( will be de fi ned later) while if k = 1 and gimel < gimel(1) where gimel(1) is the fi rst eigenvalue of the singular p - Laplacian operator given by -div [vertical bar x vertical bar(-ap) vertical bar del u vertical bar(p-2) del u], the above system has no positive weak solution. Also, we discuss the uniqueness of the positive weak solution.